L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·13-s + 14-s + 16-s + 6·17-s + 19-s + 6·23-s − 5·25-s + 2·26-s + 28-s − 6·29-s − 10·31-s + 32-s + 6·34-s + 8·37-s + 38-s + 6·41-s − 4·43-s + 6·46-s + 49-s − 5·50-s + 2·52-s + 6·53-s + 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 1.25·23-s − 25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 1.79·31-s + 0.176·32-s + 1.02·34-s + 1.31·37-s + 0.162·38-s + 0.937·41-s − 0.609·43-s + 0.884·46-s + 1/7·49-s − 0.707·50-s + 0.277·52-s + 0.824·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.231449728\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.231449728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.039804171440519140894897994877, −7.897229208863126488204431759214, −7.52444935291902841047218598997, −6.54083618156768126645931386556, −5.56998799949893065979174346107, −5.23638640702833112669407286710, −3.97866083483422929020536892791, −3.44085163590786897994899547037, −2.25567685991970798779562258398, −1.12358263831481689334600014660,
1.12358263831481689334600014660, 2.25567685991970798779562258398, 3.44085163590786897994899547037, 3.97866083483422929020536892791, 5.23638640702833112669407286710, 5.56998799949893065979174346107, 6.54083618156768126645931386556, 7.52444935291902841047218598997, 7.897229208863126488204431759214, 9.039804171440519140894897994877